Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonal faces, while all other faces are hexagons. Fullerene graphs are mathematical models of fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites and diamonds. We give a survey on fullerene graphs from our perspective, which could be also considered as an introduction to this topic. Different types of fullerene graphs are considered, their symmetries, and construction methods. We give an overview of some graph invariants that can possibly correlate with the fullerene molecule stability, such as: the bipartite edge frustration, the independence number, the saturation number, the number of perfect matchings, etc.

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a b s t r a c tFor a simple graph G with n vertices and m edges, the inequality M 1 (G)/n ≤ M 2 (G)/m, where M 1 (G) and M 2 (G) are the first and the second Zagreb indices of G, is known as the Zagreb indices inequality. A set S is good if for every graph whose degrees of vertices are in S, the inequality holds. We characterize that an interval [a, a + n] is good if and only if a ≥ n(n−1) 2 or [a, a + n] = [1, 4]. We also present an algorithm that decides if an arbitrary set S of cardinality s is good, which requires O(s 2 log s) time and O(s) space.

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